About: The improved decay rate for the heat semigroup with local magnetic field in the plane

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Attributes	Values
rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://link.springer.com/content/pdf/10.1007%2Fs00526-012-0516-1.pdf
Description	We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta. The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schrodinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behaviour of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov-Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables. Since no assumptions about the symmetry of the magnetic field are made in the present work, it gives a normwise variant of the recent pointwise results of Kovarik (Calc Var doi:10.1007/s00526-011-0437-4) about large-time asymptotics of the heat kernel of magnetic Schrodinger operators with radially symmetric field in a more general setting. We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta. The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schrodinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behaviour of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov-Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables. Since no assumptions about the symmetry of the magnetic field are made in the present work, it gives a normwise variant of the recent pointwise results of Kovarik (Calc Var doi:10.1007/s00526-011-0437-4) about large-time asymptotics of the heat kernel of magnetic Schrodinger operators with radially symmetric field in a more general setting. (en)
Title	The improved decay rate for the heat semigroup with local magnetic field in the plane The improved decay rate for the heat semigroup with local magnetic field in the plane (en)
skos:prefLabel	The improved decay rate for the heat semigroup with local magnetic field in the plane The improved decay rate for the heat semigroup with local magnetic field in the plane (en)
skos:notation	RIV/61389005:_____/13:00392916!RIV14-MSM-61389005
http://linked.open...avai/riv/aktivita	I P
http://linked.open...avai/riv/aktivity	I, P(GAP203/11/0701), P(LC06002)
http://linked.open...iv/cisloPeriodika	1-2
http://linked.open...vai/riv/dodaniDat	2014
http://linked.open...aciTvurceVysledku	Krejčiřík, David
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Ústav jaderné fyziky AV ČR, v. v. i.
http://linked.open...dnocenehoVysledku	79278
http://linked.open...ai/riv/idVysledku	RIV/61389005:_____/13:00392916
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	convection-diffusion equations; large time behavior; singular potentials; hardy inequality; dirichlet forms; operators (en)
http://linked.open.../riv/klicoveSlovo	convection-diffusion equations operators dirichlet forms hardy inequality singular potentials large time behavior
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[4F88A7057728]
http://linked.open...i/riv/nazevZdroje	Calculus of Variations and Partial Differential Equations
http://linked.open...in/vavai/riv/obor	BE
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	1 (xsd:int)
http://linked.open...vavai/riv/projekt	Guided Quantum Dynamics Doppler Institute for Mathematical Physics and Applied Mathematics
http://linked.open...UplatneniVysledku	2013
http://linked.open...v/svazekPeriodika	47
http://linked.open...iv/tvurceVysledku	Krejčiřík, David
http://linked.open...ain/vavai/riv/wos	000317970100009
issn	0944-2669
number of pages	20 (xsd:int)
http://bibframe.org/vocab/doi	10.1007/s00526-012-0516-1

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